v = λf 1. A wave is created on a Slinky such that its frequency is 2 Hz and it has a wavelength of 1.20 meters. What is the speed of this wave?

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1 Today: Questions re: HW Examples - Waves Wave Properties > Doppler Effect > Interference & Beats > Resonance Examples: v = λf 1. A wave is created on a Slinky such that its frequency is 2 Hz and it has a wavelength of 1.20 meters. What is the speed of this wave? 2. A sound wave has a wavelength of 0.80 m, and travels through air with a speed of 340 m/s. What is the frequency of this sound wave? 3. Red light has a wavelength of 6 E-7 m (600 nm, where 1 nm = 1 E-9 m). If the speed of light is taken to be ~3 E8 m/s, what is the frequency of this wave? 4. A radio wave has a frequency of 89.7 MHz (1 MHz = 10 6 Hz). What is its wavlength? (Recall that radio is an EM wave) 5. *Assume you are at a baseball game. You are 150 m from home plate. Sound travels at a speed of 340 m/s, light at 3 E 8 m/s. How much sooner will you see a ball hit than you will hear it? Doppler Effect, Interference, Resonance 1

2 Doppler Effect: The Doppler Effect describes the perceived frequency difference in waves when the wave source and observer are moving relative to one another. Imagine you are standing by the road when a police car or ambulance is approaching with its siren on. As the vehicle approaches, you notice a rise in the pitch of the siren. As the vehicle passes you and then moves away, you will hear the pitch of the siren decreasing. Consult the diagram below: Lower Pitch Higher Pitch In this figure, the police car moves from left to right, away from the woman and towards the man (the man and woman will be referred to as the observer). As the police car is driving, it is blaring its siren, which emits a sound wave with a frequency f source (this is constant) that travels with a speed v (speed of sound; this is constant; 340 m/s). The car itself is moving with a speed v source (we will assume this is also constant). As each successive sound wave is emitted, the car moves forward; what this serves to do is compress the sound waves in front of the car, and spread out the sound waves behind it. For the observer behind the car, as the waves spread out, it appears that the wavelength is getting longer (the distance between each wave increases). Doppler Effect, Interference, Resonance 2

3 By the relationship v = λf, this will cause her to interpret the sound as having a lower frequency, and hence, a lower pitch. For the observer in front of the car, as the waves are compressed, it appears that the wavelength is getting shorter (the distance between each wave decreases); this will cause him to interpret the sound as having a higher frequency, and hence, a higher pitch. Here the "Apparent Wavelength" is smaller; this implies the observer will detect a higher frequency Here the "Apparent Wavelength" is longer; this implies and observer will detect a lower frequency We can calculate the "observed" frequency using the equation where v is the speed of the wave. We choose the signs such that, when the observer and source move towards each other, the observed frequency goes up, and when they move away from each other, the observed frequency goes down. Doppler Effect, Interference, Resonance 3

4 Examples: 1. A sound wave travels through air with a speed of 340 m/s. A train moves with a speed of 40 m/s, and its horn generates a sound wave whose frequency is 300 Hz. Determine the observed frequency for a stationary observer as the train (a) approaches and (b) recedes from the observer. Wave Interference: When two waves of the same type (such as water waves, sound waves, etc.) approach and meet each other, they will combine together in a process known as Interference. There are 3 main results of this process: Constructive Interference Destructive Interference Constructive and Destructive Interference Doppler Effect, Interference, Resonance 4

5 With Constructive Interference, the crests meet up, and the troughs meet up, creating a wave with the same wavelength/ frequency, but the Amplitudes add up to create a larger wave. With Destructive Interference, the crests match up with troughs, such that they serve to eliminate each other. If the Amplitudes of both waves are the same, the wave will completely cancel. *We are assuming that these waves have the exact same wavelength, frequency and Amplitudes. If there are variances in any of these quantities, the sum of the waves is more complicated (example on previous slide). Beats: When sound waves interfere, if they have different frequencies, the sum of wave interference forms a new wave, what is known as a "Beat", and what you will hear is a series of varying soft/loud oscillations. The Beat Frequency is just the difference between the frequencies of the two waves. Soft Soft Loud Loud Loud In the diagram above, two waves (pink and green) interfere. The bottom picture shows the sum of the interference; the Sinusoidal line is referred to as the "Beat Envelope", but essentially it is just the sum of the two waves. Areas of Constructive Interference (these are called Anti-nodes) are loud, and areas of Destructive Interference (these are called Nodes) are soft. Doppler Effect, Interference, Resonance 5

6 Resonance: All objects have a Natural Frequency at which they "want" to vibrate/oscillate; this is referred to as its Resonant Frequency. When an object is forced to vibrate at its Resonant Frequency, what you find is that the Amplitude of the oscillation grows very rapidly with small inputs. Imagine pushing someone on a swing; you can get them to swing to a very large amplitude, using small pushes, as long as you push at the right time. This is what it means to force an oscillation at the resonant frequency. Image courtesy of xkcd.com When listening to a radio station in your car, the radio searches for the correct resonant frequency to match the incoming radio wave (e.g., FM). We'll now watch a few videos re: Resonance. Doppler Effect, Interference, Resonance 6